Adaptive Optical System Testbed with Karhunen-Loeve Polynomial Based Method for Simulating Atmospheric Turbulence

ABSTRACT

A system and method for simulating atmospheric turbulence for testing optical components. A time varying phase screen representing atmospheric turbulence is generated using Karhunen-Loeve polynomials and a splining technique for generating temporal functions of the noise factor for each Zernike mode. The phase screen is input to a liquid crystal spatial light modulator. A computer display allows the user to set geometric characteristics, the severity of the turbulence to be simulated, and to select between methods for generating atmospheric turbulence including Karhunen-Loeve polynomials, Zernike polynomials, and Frozen Seeing.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of U.S. patent application Ser. No.12/698,408 filed on Feb. 2, 2010, which is a nonprovisional of U.S.Provisional Application 61/149,164 filed on Feb. 2, 2009 and of U.S.Provisional Application 61/300,546, filed on Feb. 2, 2010. The entiredisclosure of each of these documents is incorporated by referenceherein.

BACKGROUND OF THE INVENTION

1. Technical Field

This application is related to systems for modeling atmosphericturbulence, and more specifically, to systems for measuring andsimulating atmospheric turbulence for adaptive optical systems.

2. Background Information

Light traveling from a star, or any point source, will propagatespherically outward. As shown in FIG. 1, after a long distance, thewavefront, or surface of equal phase, will be flat. The Earth'satmosphere is a large, non-linear, non-homogenous medium that isconstantly changing in a random fashion that affects light as itpropagates through it.

When the light begins to propagate through Earth's atmosphere, thevarying indices of refraction will alter the optical path, as shown inFIG. 2. The Earth's atmosphere is a non-homogeneous continuous turbulentmedium, and its properties vary based on factors including temperature,pressure, wind velocities, and humidity. The Earth's atmosphere alsotemporally changes in a random fashion. The Kolmogorov model foratmospheric turbulence is a description of the nature of the wavefrontperturbations introduced by the atmosphere and it is one of the mostaccepted models. It is supported by a variety of experimentalmeasurements and is quite widely used in simulations for exo-atmosphericseeing. The Kolmogorov theory of atmospheric turbulence is based on theassumption that the turbulence varies the index of refraction and thusaffects the permittivity of the medium throughout it. This affects thelight as it propagates through the atmosphere.

Adaptive Optics (AO) is a term used for a class of techniques dealingwith the correction of wavefront distortions in an optical system. Somewavefront distortions may include those caused by the atmosphere.Exo-atmospheric applications of adaptive optics particularly include thecorrection of atmospheric turbulence for a telescope system. Adaptiveoptics techniques are used in free space laser communication systems,high energy laser systems, and phase correction for deployablespace-based telescopes and imaging systems.

Prior to deployment, an adaptive optics system requires calibration andfull characterization in the laboratory. Creating realistic atmosphericsimulations has been notoriously difficult and computationallyintensive. Many techniques are currently being used with adaptive opticssystems for simulating atmospheric turbulence. One static componenttechnique uses glass phase screens with holograms etched into them.Another technique uses a rotating filter wheel with etched holographicphase screens to simulate temporal transitions. However, generating andmanufacturing these holograms can be quite expensive. Other methodssimulate atmospheric turbulence by using a static aberrator, such as aclear piece of plastic, and rotating it, or by simply using a hot-platedirectly under the beam path to cause local turbulence.

BRIEF SUMMARY OF THE INVENTION

A system and method for simulating atmospheric turbulence for testingoptical components. A time varying phase screen representing atmosphericturbulence is generated using Karhunen-Loeve polynomials and a spliningtechnique for generating temporal functions of the noise factor for eachZernike mode. The phase screen is input to a liquid crystal spatiallight modulator. A computer display allows the user to set geometriccharacteristics, and select between methods for generating atmosphericturbulence including Karhunen-Loeve polynomials, Zernike polynomials,and Frozen Seeing.

An aspect of the invention is directed to a computer implemented methodfor simulating atmospheric turbulence in an optical testbed system. Themethod includes generating a time varying wavefront representingatmospheric turbulence as input to a liquid crystal spatial lightmodulator. The wavefront calculated at each location and time as aweighted sum of Karhunen-Loeve polynomials and a temporal function ofthe noise factor generated by a spline curve fit to a sequence of randomnumbers, weighted with aberration amplitudes determined based on theZernike-Kolmogorov residual errors for each Zernike mode i. Thewavefront is output as a sequence of phase screens for the liquidcrystal spatial light modulator.

An aspect of the invention is directed to an optical testbed system forevaluating the performance of an optical component in a turbulentatmosphere. The system includes a computer processor with instructionsfor generating a time varying wavefront representing atmosphericturbulence and for inputting the time varying wavefront as a sequence ofphase screens to a liquid crystal spatial light modulator. The wavefrontis calculated at each location and time as a weighted sum ofKarhunen-Loeve polynomials and a temporal function of the noise factorgenerated by a spline curve fit to a sequence of random numbers,weighted with aberration amplitudes determined based on theZernike-Kolmogorov residual errors for each Zernike mode i. The systemalso includes a user input device for inputting user-selected variablesto the computer processor, a liquid crystal spatial light modulator, anoptional Fourier filter, an optical system to be tested, one or morewavefront sensors, and a computer display for displaying the wavefrontoutput from the test unit and the reference collimated and aberratedbeams.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates propagation of a point light source over a longdistance.

FIG. 2 illustrates distortion of a flat wavefront by the Earth'satmosphere.

FIG. 3A is an illustration of an optical system with a liquid crystalspatial light modulator in accordance with an embodiment of theinvention.

FIG. 3B illustrates the liquid crystal spatial light modulator portionof the optical system in more detail.

FIG. 4 shows a sample phase screen, which is a snapshot of thesimulation of atmospheric turbulence.

FIG. 5 shows a graphical user interface for the software that generatesthe phase screen for the spatial light modulator in the FIGS. 3A and 3Boptical system.

FIGS. 6A and 6B show a sample phase screen with secondary obscuration,and without secondary obscuration, respectively.

FIG. 7 is a table of b_(p,j) values used in calculating Karhunen-Loevepolynomials.

FIG. 8 illustrates an example vector with a few random numbers with thezero mean unitary Gaussian distribution for use in generating a noisefunction.

FIG. 9 illustrates a spline curve fit to the generated vector of FIG. 8.

FIG. 10A illustrates a phase screen generated using a Frozen Seeingtechnique, showing the aperture and direction of movement, and FIG. 10Bshows the resulting phase screen as aligned with the aperture at amoment in time.

DETAILED DESCRIPTION OF THE INVENTION

Aspects of the invention are directed to a testbed system and methodthat simulates atmospheric aberrations far more inexpensively and withgreater fidelity using a liquid crystal spatial light modulator(LC-SLM). This system allows the simulation of atmospheric seeingconditions ranging from very poor to very good and different algorithmsmay be easily employed on the device for comparison. The simulations canbe dynamically generated and modified very quickly and easily. Theatmospheric turbulence and aberration generator described herein can beused for simulating/generating atmospheric seeing conditions, as well asgenerating any aberrations needed for testing an optical system.

FIGS. 3A and 3B illustrate an optical system 10 with atmosphericturbulence generator 11 and a liquid crystal spatial light modulator 12.The liquid crystal spatial light modulator 12 receives a collimated beamfrom a light source such as a laser 14, and impresses an aberration onthe wavefront of the collimated beam 15 from the laser 14. Theaberration that is impressed on the collimated beam by the spatial lightmodulator 12 is generated by the atmospheric turbulence generator 11. Inthe exemplary embodiment described herein, the aberration is timevarying, but could also be static.

The spatial light modulator 12 modulates light in amplitude and phaseunder computer control, allowing the phase of the wavefront to be variedacross the beam.

One suitable spatial light modulator is a Holoeye spatial lightmodulator model LC2002 SLM manufactured by the Holoeye Corporation,headquartered in San Diego, Calif. Another spatial light modulator isavailable from Boulder Non-Linear Systems, headquartered in Lafayette,Colo. Other spatial light modulators can also be suitable.

The aberrated beam passes through an optional Fourier filter 13, asdiscussed further in later paragraphs, and to the optical system to betested 17. A controller inputs the control signals to the spatial lightmodulator. The controller has software that places an image on thedevice. The image is a series of black and white bands that correspondto a phase of either 0 or π. FIG. 4 shows a sample phase screen that canbe applied to the collimated light by the spatial light modulator. Thephase screen is a snapshot of the simulation of atmospheric turbulence,and has 7 or 8 fringes. The bands, or fringes, of the phase screen canalso be in grayscale, with different gray values representingmodulations between 0 and π. Grayscale phase screens are suitable foruse with spatial light modulators with higher resolution, e.g., highdefinition SLMs with resolutions of 1920×1080 pixels, or greater.

As shown in FIG. 3A, the spatial light modulator is controlled by thecomputer system with a user input device and display 18 and computer'scentral processing unit with an atmospheric turbulence generator 11programmed as instructions on the CPU.

The output of beamsplitter 21 splits the beam to measure the reference,by measuring the optical beam quality before the beam enters the LC-SLM.Beamsplitter 20 splits the beam to measure the optical beam qualityafter the beam has been aberrated by the LC-SLM. Then after goingthrough the test adaptive optics system 17, the beam quality is measuredto indicate how well the adaptive optics system 17 is performing.Wavefront sensors 22, 23, and 24 can be used to measure the beam qualityat the output of the optics system 17, and at the output of each thebeamsplitters 20 and 21. Examples of suitable wavefront sensors areShack-Hartmann wavefront sensors, phase diversity wavefront sensors,interferometers for generating interference fringes, or simply a lensand a camera to measure the focal spot or point spread function (PSF).These methods can reconstruct the phase, with the PSF equaling theintensity pattern (e.g., the absolute value squared of the Fouriertransform of the wavefront) formed from the phase (wavefront). The beamquality information from each wavefront sensor is received by the CPU 25and displayed on the second monitor 19 and/or the control computerdisplay screen 18.

FIG. 3B illustrates an exemplary configuration of the spatial lightmodulator 12 in the optical testing system of FIG. 3A in more detail.The spatial light modulator 12 is a diffractive device that can directlymodulate the phase of an incoming wavefront.

The Holoeye model LC2002 SLM can modulate the phase over a range fromzero to π (pi) radians. To utilize the full 2π radian phase modulationof the incoming wavefront, a Fourier filter 13 with either a +1 or −1diffractive order is used. The small iris in the Fourier filter allowseither the +1 or the −1 diffractive order.

The Fourier filter 13 reimages the pupil for the input of the opticalsystem under testing. The distances f₁ and f₂ are chosen to magnify ordemagnify the beam size. For a spatial light modulator that onlymodulates 0 and π, such as the LC2002SLM, when the beam is focused, theiris blocks all the diffractive orders except the +1 or −1 order toexploit the 2× multiplicative factor in phase due to diffraction fromthe spatial light modulator. By using the +1 or −1 order, the phase ismodulated by 0 to 2π.

Note that for spatial light modulators that provide 2π modulationdirectly, the Fourier filter is not necessary.

The applied electrical field changes the polarization of the liquidcrystal, changing the index of refraction n and the optical path length,and thus, the phase of the light passing through the SLM.

FIG. 5 shows an exemplary graphical user interface 30 for the controllersoftware that controls the spatial light modulator. In an exemplaryembodiment, the controller software is written in MATLAB.

The alignment of either the +1 or the −1 diffractive order through thesmall iris is done via the “tip/tilt bias for order alignment control”panel 38 in the GUI screen. Once this alignment is achieved, the usercan fine-tune the alignment of the system from aberrations caused beforeand/or after this point in the optical path through the use of the“alignment bias” panel 39 in the GUI screen. The settings on the“tip/tilt bias for order alignment control” panel 38 are a2 (tip) and a3(tilt). The settings on the alignment bias panel 39 are a1 (piston), a2(tip), a3 (tilt), a4 (focus), a5 (astig x), a6 (astig y), a7 (coma x),a8 (coma y), a9 (trefoil x), a10 (trefoil y), a11 (spherical), a12 (secastig x), and a13 (sec astig y). These settings, which correspond to thefirst 13 Zernike modes, can compensate for aberrations in the system dueto misalignments and imperfections in the system.

The software also provides the user with the option of connecting theSLM to a second computer and second monitor, so the user cansimultaneously see the input parameters, the phase screen image as itchanges over time, and the resulting output of the optical system to betested. To display the SLM information on the second monitor, the userclicks the “multiple monitors” checkbox 37.

The placement of the phase screen can be aligned via the offset panel 31by entering the vertical and horizontal values.

The user can also select between choices for determining the atmosphericturbulence generation method. The “Atmospheric turbulence generationmethod” panel 33 allows the user to choose between using the ZernikeModes method, the Karhunen-Loeve Modes method, and the Frozen Seeingmethod for simulating atmospheric turbulence.

The “Secondary Obscuration Control” panel 41 allows the user to directthe software to simulate obscurations caused by a secondary mirror andits mount. Such secondary mirrors and mounts are found in telescopes andother optical systems. FIG. 6A shows a sample phase screen withsecondary obscuration. FIG. 6B shows the same sample phase screen,without secondary obscuration. The “Include Secondary” checkbox 42enables or disables the obscuration in the simulation. The software caninclude different secondary obscurations to match different mountingsystems.

After a mask is created, the GUI displays an image of the mask 43.

Returning again to FIG. 5, simulating the atmosphere is set up using the“Atmospheric Control” panel 34. The user can set the desired values forthe primary aperture, D, and the Fried parameter, r₀. The “# of Modes”box is for controlling the number of Zernike or Karhunen-Loeve modes tobe used during simulations. Note that the “# of Modes” box will not beused for the Frozen Seeing method of generating turbulence, as discussedfurther in later paragraphs.

As another option, the user can select whether or not to include thefirst order aberrations in the simulation by checking or unchecking the“Include Piston, Tip, and Tilt” checkbox on the User Display/ControlScreen.

Once the desired values are entered, the user can select the “Set SLM”option 32, to update the device settings.

The example Holoeye liquid crystal spatial light modulator has 800×600pixels, and can accept a beam of 21 mm in diameter. Control of the SLMcan be provided via the computer's VGA (video graphics array) or DVI(digital video interface) port and the computer's graphics card.

The following discussion provides information related to the Zernike,Karhunen-Loeve, and Frozen Seeing methods for simulating atmosphericturbulence.

For a circular aperture, a generalized pupil function can be written as

(x,y)=P(x,y)e ^(j(2π/λ)W(x,y))  Eq. (1)

where P(x, y) is the circular function circ(ρ), λ is the wavelength, andW(x, y) is the effective length error, or error in the wavefront. It isthis wavefront error, W(x, y), that atmospheric turbulence andaberrations induce and degrade the image quality of an optical imagingsystem. Aberrations can be introduced into an optical system with theliquid crystal spatial light modulator.

The Kolmogorov model describes the nature of wavefront perturbationsintroduced by the Earth's atmosphere on light. The Earth's atmosphere isa large, non-linear, non-homogeneous, constantly changing medium thataffects the light passing through it. The Kolmogorov model ofatmospheric turbulence uses the assumption that the turbulence in theatmosphere varies the index of refraction, and thus, the permittivity ofthe atmosphere for light or electric fields propagating through theatmosphere.

The wavefronts to be applied to the beam can be determined based onZernike polynomials, Karhunen-Loeve polynomials, or a frozen seeingmethod.

Zernike Polynomial Method

The lowest order Zernike polynomial is a uniform phase change across awavefront, known as piston. The next-lower-order aberration is a pairknown as tip and tilt, which are angular changes in the wavefront.Lower-order Zernike polynomials can be thought of as smoothly varying inphase across a wavefront. As the Zernike order increases, the phasevariations over the wavefront increase on smaller spatial scales.

The following table illustrates a series of Zernike polynomials.

n |m| 1 2 3 4 0 2rsin θ 2rcos θ 1 1.73 4r²(2r² − 1) 2.83(3r² − 2r)sin θ2.236(6r² − 6r + 1) 2.83(3r² − 2r)cos θ 2 2.4r²cos 2θ 3.162(4r² −3r²)sin θ 2.4r²sin 2θ 3.162(4r² − 3r²)cos θ

The computer processor can determine the phase variations to be appliedto the beam according to the following equation based on Zernikepolynomials:

$\begin{matrix}{{{Wavefront}\left( {\rho,\theta} \right)} = {\sum\limits_{i}{\left( {1 + X_{i}} \right)a_{i}{Z_{i}\left( {\rho,\theta} \right)}}}} & {{Eq}.\mspace{14mu} (2)}\end{matrix}$

In this equation, “Wavefront” corresponds to the W(x, y) term inEquation (1) above, the Zernike modes are denoted by the index i, theX_(i) values are the noise factors for the Zernike modes based on azero-mean unitary Gaussian random distribution, and the a_(i) values arethe amplitudes of the aberrations for each Zernike mode.

The Zernike polynomials Z_(i)(ρ,θ) are given by:

Z _(i)(ρ,θ)=√{square root over (n+1)}R _(n) ^(m)(ρ)cos(θ) for m≠0 and iis even;

Z _(i)(ρ,θ)=√{square root over (n+1)}R _(n) ^(m)(ρ)sin(θ) for m≠0 and iis odd; and

Z _(i)(ρ,θ)=R _(n) ⁰(ρ) for m=0,  Eq. (3)

where

$\begin{matrix}{{R_{n}^{m}(\rho)} = {\sum\limits_{s = 0}^{{({n - m})}/2}\left\lbrack {\frac{\left( {- 1} \right)^{s}{\left( {n - s} \right)!}}{{{s\left( {\frac{n + m}{2} - s} \right)}!}{\left( {\frac{n - m}{2} - s} \right)!}}\rho^{n - {2\; s}}} \right\rbrack}} & {{Eq}.\mspace{14mu} (4)}\end{matrix}$

The azimuthal and radial orders of the Zernike polynomials, m and n,respectively, satisfy the conditions that m≦n and n−m=even, and i is theZernike order number.

The Zernike polynomials have the advantage that the first few termsresemble the classical aberrations known to lens makers. The Zernikeorder number is related to the azimuthal and radial numbers via thenumerical pattern in the following table. The term for piston, a_(l), isneglected as it is just a constant bias in the overall phase.

i n m 1 0 0 2 1 1 3 1 −1 4 2 0 5 2 2 6 2 −2 7 3 1 8 3 −1 9 3 3 10 3 −311 4 0 12 4 2 13 4 −2 14 4 4 15 4 −4 16 5 1 17 5 −1 18 5 3 19 5 −3 20 55 21 5 −5 22 6 0 23 6 2 24 6 −2 25 6 4 26 6 −4 27 6 6 28 6 −6

The amplitudes a_(i) of Equation (2) are the coefficients of each of theZernike modes, are integers, and are calculated based on theZernike-Kolmogorov residual errors (Δ_(j)). The amplitude of theaberration a_(i) can be found as shown in Robert J. Noll, “Zernikepolynomials and atmospheric turbulence”, J. Opt. Soc. Am., Vol. 66, No.3, March 1976, incorporated herein in its entirety. Noll calculatedthese Zernike coefficients a_(i) statistically and measured themexperimentally. These coefficients are proportional to the residualerror Δ's, which are related to the diameter of the aperture, D, and theFried parameter, r₀.

As an example, the following table shows the values of a_(i) for D/r₀equal to 2.25 and the number of modes M=21:

i a_(i) 2 1.3423 3 0.3844 4 −0.4742 5 −1.5124 6 0.3981 7 −0.2772 8−0.4476 9 −0.2197 10 0.2499 11 −0.0670 12 0.2672 13 0.3445 14 −0.0917 150.0451 16 0.1798 17 0.0035 18 −0.0103 19 −0.1457 20 0.0522 21 0.1213

The Zernike modes and corresponding Zernike-Kolmogorov residual errorsare shown in the table below.

Zernike-Kolmogorov residual Zernike mode error Tip Δ₁ =1.0299(D/r₀)^(5/3) Tilt Δ₂ = 0.5820(D/r₀)^(5/3) Focus Δ₃ =0.1340(D/r₀)^(5/3) Astigmatism X Δ₄ = 0.0111(D/r₀)^(5/3) Astigmatism YΔ₅ = 0.0880(D/r₀)^(5/3) Coma X Δ₆ = 0.0648(D/r₀)^(5/3) Coma Y Δ₇ =0.0587(D/r₀)^(5/3) Trefoil X Δ₈ = 0.0525(D/r₀)^(5/3) Trefoil Y Δ₉ =0.0463(D/r₀)^(5/3) Spherical Δ₁₀ = 0.0401(D/r₀)^(5/3) SecondaryAstigmatism X Δ₁₁ = 0.0377(D/r₀)^(5/3) Secondary Astigmatism Y Δ₁₂ =0.0352(D/r₀)^(5/3) Larger Orders (J > 12) Δ_(J) = ~0.02944J{square rootover (^(3/2))}(D/r₀)^(5/3)

The r₀ term in the table above is the Fried parameter, which is theeffective size of the atmospheric turbulence cells. The D term is theaperture diameter. The ratio of r₀/D is important in determining whetherimprovements in image quality can be achieved by compensating imagemotion. Additional information related to the r₀/D ratio is discussed in“Introduction to Image Stabilization”, Scott Teare and Sergio R.Restaino, Tutorial Texts in Optical Engineering, Vol. TT73, SPIE Press,2006, incorporated herein in its entirety.

Thus, the wavefront for a particular radius and angular location (ρ,θ)is a weighted sum of the noise multiplied by the amplitude of theaberration a_(i) and the Zernike polynomial Z_(i) (ρ,θ).

Time dependence of the generated wavefront in Equation (2) can beincluded as follows.

Phase variances can be described as having Gaussian random distribution,according to Tatarskii's model, in V. I. Tatarskii, Wave Propagation ina Turbulent Medium, McGraw-Hill, 1961.

Equation (2) can be written as

$\begin{matrix}{{{Wavefront}\left( {\rho,\theta} \right)} = {\sum\limits_{i}{\left( {1 + {X_{i}(t)}} \right)a_{i}{Z_{i}\left( {\rho,\theta} \right)}}}} & {{Eq}.\mspace{14mu} (5)}\end{matrix}$

Here, X_(i)(t) are the temporal functions of the noise factor for theith mode based on a zero-mean Gaussian random distribution. A vector ofX_(i) values is generated with a few random numbers with the zero meanunitary Gaussian distribution. The X_(i)'s can be modified from beingjust random numbers, to a continuous function of time for each mode.

A spline curve, or “cubic spline data interpolation” curve, is fit tothe generated vector of the few random numbers (X_(i)'s) to produce theX_(i)(t) temporal function. The spline curve can be generated using asoftware package written in a commercially available software languagesuch as MATLAB, or another computer programming language.

The software instructions include a default value for the length of timeover which the simulation should be run, for example, 20 minutes,although a shorter or longer simulation time can be used. The softwaredetermines the number of random numbers Xi(t) to be generated based onthe length of time for the simulation.

Karhunen-Loeve Polynomial Method

Alternatively, the computer processor can generate the wavefront usingKarhunen-Loeve polynomials. The Karhunen-Loeve polynomials arestatistically independent set of orthonormal polynomials. The wavefrontcan be generated according to the equation:

$\begin{matrix}{{{Wavefront}(t)} = {\sum\limits_{i}{\left( {1 + {X_{i}(t)}} \right)a_{i}{K_{i}\left( {\rho,\theta} \right)}}}} & {{Eq}.\mspace{14mu} (6)}\end{matrix}$

The K_(i)(ρ,θ) values are the Karhunen-Loeve polynomials, and are givenby:

K _(i)(ρ,θ)=Σ_(j=1) ^(N) [b _(p,j) Z _(j)(ρ,θ)]  Eq. (7)

where the b_(p,j) values are as calculated by Wang, J. Y. & Markey, J.K. (1978), “Modal compensation of atmospheric turbulence phasedistortion,” J. Opt. Soc. Am. 68, pp. 78-87, and as shown in FIG. 7. Thevalue of N is the number of Zernike orders in which the Karhunen-Loeveorder j is represented, truncated to a finite number for computation.The elements of the summation of Eq. (6) become small after a few jterms. To represent a wavefront, Equation (6) can be written as

Wavefront(ρ,θ)=Σ_(i=1) ^(M) a _(i) K _(i)(ρ,θ)  Eq. (8)

Here, M is the number of modes to be used in the simulation, shown inthe Atmospheric Control box 34 on the GUI of FIG. 5.

The aberration a_(i) values are as described in the Zernike methodabove.

Here, the X_(i)(t) values are temporal functions of the noise factor forthe ith mode based on a zero-mean Gaussian random distribution. TheX_(i)(t) function in Equation (6) can be generated using random numbers,as follows.

FIG. 8 illustrates an example vector with a few random numbers with thezero mean unitary Gaussian distribution.

Phase variances can be described as having Gaussian random distribution,according to Tatarskii's model, in V. I. Tatarskii, Wave Propagation ina Turbulent Medium, McGraw-Hill, 1961.

Equation (5) can be written as

$\begin{matrix}{{{Wavefront}\left( {\rho,\theta} \right)} = {\sum\limits_{i}{\left( {1 + X_{i}} \right)a_{i}{K_{i}\left( {\rho,\theta} \right)}}}} & {{Eq}.\mspace{14mu} (9)}\end{matrix}$

where X_(i) is the amount of noise for a mode i based on a zero-meanunitary Gaussian random distribution.

The X_(i)'s can be modified from being just random numbers, as shown inFIG. 8, to a continuous function of time for each mode. Equation (9) canbe written as

$\begin{matrix}{{{Wavefront}\left( {\rho,\theta} \right)} = {\underset{i = 1}{\sum\limits^{M}}{\left( {1 + {X_{i}(t)}} \right)a_{i}{K_{i}\left( {\rho,\theta} \right)}}}} & {{Eq}.\mspace{14mu} (10)}\end{matrix}$

A spline curve, or “cubic spline data interpolation” curve, is fit tothe generated vector of a few random numbers (X_(i)'s) to produce theX_(i)(t) temporal function. The spline curve can be generated using asoftware package written in a commercially available software languagesuch as MATLAB, or another computer programming language. FIG. 9 showsthe X_(i)(t) as the spline curve fit to the random numbers of FIG. 8.

Note that one advantage of using a continuous function such as the spinecurve is that the temporal transition of the wavefronts in theatmospheric turbulence simulation is continuous and smooth, bettersimulating the Earth's atmosphere as a continuous medium. Without usingthe splining technique, the change between phase screens would bediscontinuous and would produce a less accurate representation of theatmosphere for testing the adaptive optics system.

The software instructions include a default value for the length of timeover which the simulation should be run, for example, 20 minutes,although a shorter or longer simulation time can be used. The softwaredetermines the number of random numbers (X_(i)) to be generated based onthe length of time for the simulation.

Frozen Seeing Method

The user interface of FIG. 4 allows the user to select the Frozen Seeingmethod, based on the Taylor approximation, as the method of simulatingtemporal effects of atmospheric turbulence.

The Taylor approximation assumes that given a realization of a largeportion of atmosphere, it drifts across the aperture of interest with aconstant velocity determined by local wind conditions, but without anyother change whatsoever. For example, a large holographic phase screencan be generated and simply moved across an aperture. This technique hasproven to be a good approximation given the limited capabilities ofsimulating accurate turbulence conditions in a laboratory environment.

FIG. 10A illustrates a phase screen, showing the aperture and directionof movement. The subsection of that phase screen image aligned with theaperture represents the atmosphere at a moment in time. An example isshown in FIG. 10B. As the phase screen image is moved across theaperture, the next portion of the phase screen image that is alignedwith the aperture represents the next moment in time. This process isrepeated until the edge of the N×N phase screen is reached.

When the Frozen Seeing method is selected by the user, the phase screencan be generated by the computer using the Kolmogorov model, or othersuitable methods.

The technique can be modified to move the phase screen in a non-linearfashion across the aperture, for example, in a circular motion, however,this can lead to an atmospheric simulation that is very repetitious.

The Frozen Seeing method is far more computationally complex than theKarhunen-Loeve or Zernike polynomial methods described above. Inaddition, a larger phase screen can provide a longer simulation time,however, generating a larger phase screen can be computationallyintensive. For example, generating a 3000×3000 phase screen ofatmosphere for the Frozen Seeing method in Matlab on a Dual Core AMDTurion X2 Ultra x64 2.2 GHz processor with 4 GB of RAM takesapproximately 18 seconds. Generating a 7000×7000 phase screen takesapproximately 120 seconds. Using the spline technique with theKarhunen-Loeve polynomials can produce a realistic simulation ofatmospheric turbulence that lasts for longer periods of time with farless computational requirements.

The Frozen Seeing option does not require or allow the user to use someof the GUI options. For example, the user does not allow the user toinput a number of modes.

Note that the color or grayscale phase screen of FIGS. 10A and 10B canbe converted to a 0-π-2π phase screen with only black and white fringesfor application to a relatively low resolution LC-SLM systems.Alternatively, grayscale phase screens can be applied directly toLC-SLMs with higher resolution.

In each of the Zernike, Karhunen-Loeve, and Frozen Seeing methods, thesoftware generates an atmospheric turbulence simulation. The phasescreen generated by the software also includes modifications due to theuser directed geometrical settings including tip/tilt bias and alignmentbias, and due to secondary obscuration, if this feature is selected bythe user. The software provides an electrical input to the LC-SLM, whichspatially modulates the phase of the light passing through the LC-SLMaperture according to the phase screen.

For each of these methods, the resolution of the phase screen to begenerated is set in the software for a particular model of spatial lightmodulator. The location and resolution for calculating the wavefrontaberrations at the (ρ,θ) values should correspond to the pixels of thespatial light modulator. This setting can be modified for differentpixel resolutions of different spatial light modulators. For example,the Holoeye LC2002 has 800×600 pixels, a BNS spatial light modulator has1280×1024 pixels, and a Holoeye HD SLMs has 1920×1080 pixels. Differentversions of software can be stored for each model so that the rightresolution is used for the corresponding device. Alternatively, thesoftware can allow the user to select the model of SLM and the softwarecan adjust the number and location of the (ρ,θ) points accordingly.

Computer implementation of the methods described herein includessoftware programs with machine readable instructions for accomplishingeach task. The instructions can be included in computer readable media.Hardware systems such as free space laser systems, earth or sea basedtelescopes, deployable space based telescopes, imaging systems includinglenses and adaptive optics systems, can include modules includingatmospheric turbulence simulators as described herein.

Embodiments of the invention also include methods for testing an opticalsystem, methods for generating phase screens with a LC-SLM, and softwareand computer programs for implementing these methods using thetechniques described herein.

Embodiments of the invention also include transforming user input to thecomputer system graphical user interface into a temporally changingoptical wavefront using the LC-SLM.

Embodiments of the present invention may be described in the generalcontext of computer code or machine-usable instructions, includingcomputer-executable instructions such as program modules, being executedby a computer or other machine, such as a personal data assistant orother handheld device. Generally, program modules including routines,programs, objects, components, data structures, and the like, refer tocode that performs particular tasks or implements particular abstractdata types. Embodiments of the invention may be practiced in a varietyof system configurations, including, but not limited to, handhelddevices, consumer electronics, general purpose computers, specialtycomputing devices, and the like. Embodiments of the invention may alsobe practiced in distributed computing environments where tasks areperformed by remote processing devices that are linked through acommunications network. In a distributed computing environment, programmodules may be located in association with both local and remotecomputer storage media including memory storage devices. The computeruseable instructions form an interface to allow a computer to reactaccording to a source of input. The instructions cooperate with othercode segments to initiate a variety of tasks in response to datareceived in conjunction with the source of the received data.

Computing devices includes a bus that directly or indirectly couples thefollowing elements: memory, one or more processors, one or morepresentation components, input/output (I/O) ports, I/O components, andan illustrative power supply. Bus represents what may be one or morebusses (such as an address bus, data bus, or combination thereof). Onemay consider a presentation component such as a display device to be anI/O component. Also, processors have memory. Categories such as“workstation,” “server,” “laptop,” “hand held device,” etc., as all arecontemplated within the scope of the term “computing device.”

Computing device typically includes a variety of computer-readablemedia. By way of example, and not limitation, computer-readable mediamay comprise Random Access Memory (RAM); Read Only Memory (ROM);Electronically Erasable Programmable Read Only Memory (EEPROM); flashmemory or other memory technologies; CDROM, digital versatile disks(DVD) or other optical or holographic media; magnetic cassettes,magnetic tape, magnetic disk storage or other magnetic storage devices,or any other tangible physical medium that can be used to encode desiredinformation and be accessed by computing device.

Memory includes computer storage media in the form of volatile and/ornonvolatile memory. The memory may be removable, nonremovable, or acombination thereof. Exemplary hardware devices include solid statememory, hard drives, optical disc drives, and the like. Computing deviceincludes one or more processors that read from various entities such asmemory or I/O components. Presentation component can present dataindications to a user or other device. Exemplary presentation componentsinclude a display device, speaker, printing component, vibratingcomponent, and the like.

I/O ports allow computing device to be logically coupled to otherdevices including I/O components, some of which may be built in.Illustrative components include a microphone, joystick, game pad,satellite dish, scanner, printer, wireless device, etc.

The example shown in this application uses MATLAB software, althoughother software languages or systems are contemplated within theinvention.

Obviously, many modifications and variations of the present inventionare possible in light of the above teachings. It is therefore to beunderstood that the claimed invention may be practiced otherwise than asspecifically described.

What is claimed as new and desired to be protected by Letters Patent ofthe United States is:
 1. An optical testbed system for evaluating theperformance of an optical component in a turbulent atmosphere, thesystem comprising: a computer processor with instructions for generatinga time varying wavefront representing atmospheric turbulence and forinputting the time varying wavefront as a sequence of phase screens to aliquid crystal spatial light modulator; a display screen operablyconnected to the computer processor, wherein in operation, the displayscreen displays a plurality of alternative methods of generating eachphase screen, the methods including a Karhunen-Loeve polynomial basedmethod, a Zernike polynomial based method, and a frozen seeing method;and a user input device for receiving from a user and inputting to thecomputer processor a user-selected one of the methods.
 2. The opticaltestbed system according to claim 1, said display further adapted todisplay a Fried parameter r₀, the Fried parameter representing theseverity level of turbulence to be simulated, and said user input devicefurther configured to receive from a user and to input to the computerprocessor a user-selected value of the Fried parameter.
 3. The opticaltestbed system according to claim 2, wherein the frozen seeing methodincludes initially generating a phase screen larger in size than anaperture of the optical system, and sequentially applying subsections ofthe phase screen to the spatial light modulator to simulate atmosphericturbulence.
 4. The optical testbed system according to claim 1, whereinthe Karhunen-Loeve polynomial based method includes: calculating thewavefront at each location and time as sum over a plurality of Zernikemodes of a product of Karhunen-Loeve polynomials, a temporal function ofthe noise factor, and aberration amplitudes determined based on theZernike-Kolmogorov residual errors for each Zernike mode.
 5. The opticaltestbed system according to claim 1, wherein the Karhunen-Loevepolynomial based method includes generating the wavefront according to:${{{Wavefront}\left( {\rho,\theta} \right)} = {\sum\limits_{i}{\left( {1 + {X_{i}(t)}} \right)a_{i}{K_{i}\left( {\rho,\theta} \right)}}}},$wherein for each mode i, a_(i) is the amplitude of the aberrationdetermined based on the Zernike-Kolmogorov residual errors, K_(i)(ρ,θ)is the Karhunen-Loeve polynomial, and (1+X_(i)(t)) is the temporalfunction of the noise factor.
 6. The optical testbed according to claim1, further comprising: the spatial light modulator, configured toreceive a collimated light beam, impress the wavefront on the collimatedlight beam, and output a resulting aberrated beam to an opticalcomponent to be tested.
 7. The optical testbed system according to claim6, wherein the liquid crystal spatial light modulator has a phasemodulation range of 0 to π radians, and the system further comprises: aFourier filter arranged optically between the spatial light modulatorand the optical component to be tested.
 8. The system according to claim7, wherein the optical component is a telescope.
 9. The optical testbedsystem according to claim 1, wherein the computer processor includesinstructions for adding secondary obscuration to the phase screen. 10.An optical testbed system for evaluating the performance of an opticalcomponent in a turbulent atmosphere, the system comprising: a computerprocessor with instructions for generating a time varying wavefrontrepresenting atmospheric turbulence and for inputting the time varyingwavefront as a sequence of phase screens to a liquid crystal spatiallight modulator; a display screen operably connected to the computerprocessor, wherein in operation, the display screen displays a pluralityof alternative methods of generating each phase screen, the methodsincluding a Karhunen-Loeve polynomial based method and at least oneother method; and a user input device for receiving from a user andinputting to the computer processor a user-selected one of the methods.